Discrete Optimization
A variable neighborhood search with an effective local search for uncapacitated multilevel lot-sizing problems

https://doi.org/10.1016/j.ejor.2013.10.025Get rights and content

Highlights

  • A new VNS algorithm for solving uncapacitated multilevel lot-sizing problems is developed.

  • A local search called the Predecessors Depth-first Traversal Search is embedded in VNS.

  • An approach for fast calculation of the cost change for the VNS algorithm is proposed.

  • The experimental results show that the new VNS algorithm outperforms all of the existing algorithms.

Abstract

In this study, we improved the variable neighborhood search (VNS) algorithm for solving uncapacitated multilevel lot-sizing (MLLS) problems. The improvement is twofold. First, we developed an effective local search method known as the Ancestors Depth-first Traversal Search (ADTS), which can be embedded in the VNS to significantly improve the solution quality. Second, we proposed a common and efficient approach for the rapid calculation of the cost change for the VNS and other generate-and-test algorithms. The new VNS algorithm was tested against 176 benchmark problems of different scales (small, medium, and large). The experimental results show that the new VNS algorithm outperforms all of the existing algorithms in the literature for solving uncapacitated MLLS problems because it was able to find all optimal solutions (100%) for 96 small-sized problems and new best-known solutions for 5 of 40 medium-sized problems and for 30 of 40 large-sized problems.

Introduction

The multilevel lot-sizing (MLLS) problem addresses how to best determine the trade-off cost in a production system with the purpose of satisfying customer demand with a minimum total cost. The MLLS problem plays an important role in modern production systems of manufacturing and assembly firms. Many planning systems, such as Material Request Planning (MRP) and Master Product Scheduling (MPS), depend heavily on the basic mathematical model and solution approaches for the MLLS problem. Nevertheless, the MLLS problem was proven to be strongly NP-hard (Arkin, Joneja, & Roundy, 1989). Optimal solutions to large-sized problems with complex product structures are notably difficult to find. In addition, optimal algorithms exist only for small-sized MLLS problems, and these algorithms include dynamic programming formulations (Zangwill, 1968, Zangwill, 1969), an assembly–structure-based method (Crowston & Wagner, 1973), and branch-and-bound algorithms (Afentakis and Gavish, 1986, Afentakis et al., 1984). Many heuristic approaches have been developed to solve the MLLS problem and its variants with near-optimal solutions. Early studies first applied sequential applications of single-level lot-sizing models to each component of the product structure (Veral and LaForge, 1985, Yelle, 1979), and later studies used an approximate application of multilevel lot-sizing models (Blackburn and Millen, 1982, Blackburn and Millen, 1985, Coleman and McKnew, 1991).

The uncapacitated MLLS acts as a fundamental problem, and its solution approach could be highly meaningful to many of its extended versions, including the capacitated MLLS, the MLLS with time-windowing, and the MLLS with order acceptance. In practice, many SME firms in China’s electromechanical industry are more willing to adopt dynamic capability policies because they can improve their capacities during busy seasons with many methods, such as extra working-time, temporal employment, and rented machines. Therefore, the uncapacitated MLLS model caters to the situations of their ERP systems.

Over the past decade, several metaheuristic algorithms have been developed to solve uncapacitated MLLS problems with complex product structures. It has been reported that these algorithms are capable of providing highly cost-efficient solutions with a reasonable computing load. Dellaert and Jeunet (2000) and Dellaert, Jeunet, and Jonard (2000) first presented a hybrid genetic algorithm (HGA) for solving uncapacitated MLLS problems with a general product structure and introduced a competitive strategy for mixing the use of five operators in the evolution of the chromosomes from one generation to the next. Homberger (2008) presented a parallel genetic algorithm (PGA) and an empirical policy for deme migration (rate, interval, and selection) for the MLLS problem. These researchers used the power of parallel calculations to decentralize the large calculation load over multiple processors (30 processors were used in their experiments). In addition to genetic algorithms, other metaheuristic algorithms, such as simulated annealing (SA) algorithms (Homberger, 2010, Tang, 2004), the particle swarm optimization (PSO) algorithm by Han, Tang, Kaku, and Mu (2009), the MAX–MIN ant colony optimization (ACO) system by Pitakaso et al., 2006, Pitakaso et al., 2007, and the soft optimization approach (SOA) based on segmentation by Kaku and Xu, 2006, Kaku et al., 2010, have been developed for solving uncapacitated MLLS problems.

The variable neighborhood search (VNS) algorithm initiated by Mladenovic and Hansen (1997) is a top-level methodology for solving optimization problems. Because its principle is simple and easy to understand and implement, various VNS-based algorithms have been successfully applied to many optimization problems (Hansen et al., 2008a, Hansen et al., 2008b, Hansen et al., 2010). Mladenovic, Urosevic, Hanafi, and Ilic (2012) presented a new schema of the general variable neighborhood search (GVNS), which is an extended version of the basic VNS that considers multiple neighborhood structures. Labadie, Mansini, Melechovsky, and Calvo (2012) proposed a VNS procedure based on the idea of exploring, most of the time, granular instead of complete neighborhoods in order to improve the algorithms efficiency without loosing effectives. A brief summarization of recent successful VNS applications can be found in Mladenovic, Kratica, Kovacevic-Vujcic, and Cangalovic (2012).

Xiao, Kaku, Zhao, and Zhang (2011a) first developed a VNS-based algorithm for basic schema and a shift rule to solve small- and medium-sized MLLS problems; this algorithm performed better than the HGA in small- and medium-sized problems. Xiao, Kaku, Zhao, and Zhang (2011b) developed a reduced VNS (RVNS) combined with six SHAKING techniques to solve large-sized MLLS problems. The term “reduced” indicates a simplified version of the classical VNS algorithm because the local search (the most time-consuming component of VNS) was removed from the basic scheme. Although RVNS is still a generate-and-test algorithm, it differs significantly from the single-point stochastic search (SPSS) algorithm (Jeunet & Jonard, 2005) because it uses a systematic method to change multiple bits (with a maximum Kmax) in the incumbent to generate a candidate, whereas the latter changes only a single bit. In the study conducted by Xiao, Kaku, Zhao, and Zhang (2012), three indices (i.e., the distance, changing range, and changing level) were proposed for a neighborhood search based on which three hypotheses were verified and can be used as common rules to enhance the performance of any existing generate-and-test algorithm. Using these three hypotheses, the proposed iterated neighborhood search (INS) algorithm delivered notably good performance when tested against 176 benchmark problems.

In our previous research, the neighborhood structure is defined under a distance-based metric that measures the distance (of two solutions) using the number of different bits, and another type of neighborhood structure based on problem decomposition was also studied in the recent literature (Helber and Sahling, 2010, Lang and Shen, 2011, Seeanner et al., 2013). Several decomposition methods (i.e., product-oriented, time-oriented, and resource-oriented) combined with fix-and-optimized (or partial optimization) strategies were adapted to decompose the original problem into multiple sub-problems in order to restrain the optimization to a smaller area of the binary variables.

In this paper, we developed an effective local search procedure known as the Ancestors Depth-first Traversal Search (ADTS) for the RVNS algorithm such that the RVNS algorithm can be restored to a standard VNS. Although the ADTS procedure adds a considerable amount of computing load to the algorithm, we successfully developed an efficient method (known as trigger) using a new formulation of MLLS problems to rapidly calculate the change in the objective cost during the neighborhood search process. Thus, the new VNS algorithm is both effective and efficient in solving MLLS problems with high-quality solutions and within an acceptable computing time.

The remainder of this paper is organized as follows. In Section 2, we describe the new formulation of the MLLS problem. In Section 3, we detail a local search procedure known as ADTS which was added to our previously presented RVNS algorithm for effectively solving the MLLS problem. Section 4 outlines a highly efficient approach for rapidly calculating the cost variation of the objective function as the incumbent solution changes. In Section 5, we test the proposed algorithm on 176 benchmark MLLS problem instances of different scales (small, medium, and large) and compare its performance with that of existing methods. Finally, Section 6 presents our concluding remarks.

Section snippets

Problem formulation and neighborhood definition

The MLLS problem under investigation is considered an uncapacitated, discrete-time, multilevel production/inventory system with a general product structure1 and multiple-end items. We assume that external demands for the end items are known throughout the planning horizon and

The mechanism of VNS for MLLS problems

The VNS is a top-level heuristic strategy that can be used to conduct an efficient stochastic search for better solutions using predefined and systematic changes of neighborhoods. This approach implies a general principle of from near to far for exploration of the neighborhoods of the incumbent solution. The underlying basis for this principle is that, in most cases, there is always a higher probability of finding better solutions in nearby neighborhoods rather than farther ones. Therefore, the

An efficient method for quickly calculating the cost variation

Although the ADTS local search is quite effective in searching for better solutions, it is also a notably time-consuming procedure because it must recalculate the objective cost of the incumbent solution at each step of the recursive path (in theory). To improve the computational load, we propose an efficient method to calculate the cost variation of a solution change using the advantages of the new formulation for the MLLS problem shown in Eqs. (1), (2), (3), (4), (5). The new approach is a

Problem instances and test environment

Three classes of uncapacitated MLLS problems of different scales (small, medium, and large) were used to test the performance of the new VNS algorithm with the ADTS local search. Class 1 consists of 96 small-sized MLLS problems involving a five-item assembly structure over a 12-period planning horizon, as developed by Coleman and McKnew (1991) based on the work of Veral and LaForge (1985) and Benton and Srivastava (1985). Class 2 consists of 40 medium-sized MLLS problems involving 40/50-item

Conclusions

The uncapacitated MLLS model represents a basic form of many extended versions of the MLLS problem under various constraints. The solution approach is foundational and plays a basic role in the solution of many other extended problems. We recently conducted studies on this problem using a variable neighborhood search and presented certain effective and efficient techniques and algorithms. In this paper, we continue our contributions to this topic and focus primarily on two points. First, we

Acknowledgements

This work was supported by the National Natural Science Foundation of China under Projects No. 71271009, 71271010, and 71071007 and the Japan Society of Promotion of Science (JSPS) under Grant No. 24510192.

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